Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-05T21:18:46.159Z Has data issue: false hasContentIssue false

On dividing real algebraic varieties

Published online by Cambridge University Press:  01 March 1998

J. BOCHNAK
Affiliation:
Department of Mathematics, Vrije Universiteit, De Boelelaan 1081, 1081 HV - Amsterdam, The Netherlands; e-mail: [email protected]
W. KUCHARZ
Affiliation:
Department of Mathematics, University of New Mexico, Albuquerque, New Mexico 87131, U.S.A.; e-mail: [email protected]

Abstract

Every nonsingular projective real algebraic curve C has a unique, up to isomorphism over ℝ, nonsingular projective complexification V. If V\C is disconnected, then C is said to be dividing (we identify C with the set of real points of V). Classical results supply numerous examples of dividing real curves. This class of curves was first studied by Felix Klein [11]. A characterization of dividing real curves is given in [6].

In higher dimensions the situation is more complicated. First of all, every nonsingular projective real algebraic variety X of dimension d always has several nonisomorphic nonsingular projective complexifications, provided that d[ges ]2. Furthermore, if d[ges ]2 and W is a nonsingular projective complexification of X, then W\X is connected (we identify X with the set of real points of W). What does it then mean for X to be dividing? An answer can be given in terms of homology theory. Let K be a principal ideal domain. Assume that X, regarded as a topological manifold, is orientable over K. We say that X is dividing over K if there exists a fundamental homology class of X over K whose image in Hd(W, K) under the homomorphism induced by the inclusion map X[rarrhk ]W is zero. In the present paper we prove that this definition does not depend on the choice of W, and give a characterization of real varieties dividing over ℚ. For more information on real varieties dividing over ℤ/2 the reader may consult [18] (please note that terminology in [18] is different than here). We also discuss the relationship between real varieties dividing over ℚ (or ℤ) and dividing over ℤ/2. It follows from well-known facts that a real curve is dividing over K if and only if it is dividing.

Type
Research Article
Copyright
Cambridge Philosophical Society 1998

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)